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Miller effect - Wikipedia, the free encyclopedia
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Miller effect
From Wikipedia, the free encyclopedia
In electronics, the Miller effect accounts for the increase in the equivalent input capacitance of an
inverting voltage amplifier due to amplification of the effect of capacitance between the input and
output terminals. The virtually increased input capacitance due to the Miller effect is given by
where
is the gain of the amplifier and C is the feedback capacitance.
Although the term Miller effect normally refers to capacitance, any impedance connected between
the input and another node exhibiting gain can modify the amplifier input impedance via this effect.
These properties of the Miller effect are generalized in the Miller theorem.
Contents
■ 1 History
■ 2 Derivation
■ 3 Effects
■ 3.1 Mitigation
■ 4 Impact on frequency response
■ 4.1 Miller approximation
■ 5 References and notes
■ 6 See also
History
The Miller effect was named after John Milton Miller.[1] When Miller published his work in 1920, he
was working on vacuum tube triodes; however, the same theory applies to more modern devices
such as bipolar and MOS transistors.
Derivation
Consider an ideal inverting voltage amplifier of
gain
with an impedance connected
between its input and output nodes. The output
voltage is therefore
. Assuming
that the amplifier input draws no current, all of
the input current flows through , and is
therefore given by
.
The input impedance of the circuit is
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An ideal voltage inverting amplifier with an
impedance connecting output to input.
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Miller effect - Wikipedia, the free encyclopedia
If Z represents a capacitor with impedance
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, the resulting input impedance is
.[2]
Thus the effective or Miller capacitance CM is the physical C multiplied by the factor
Effects
As most amplifiers are inverting (
as defined above is positive), the effective capacitance at their
inputs is increased due to the Miller effect. This can reduce the bandwidth of the amplifier,
restricting its range of operation to lower frequencies. The tiny junction and stray capacitances
between the base and collector terminals of a Darlington transistor, for example, may be drastically
increased by the Miller effects due to its high gain, lowering the high frequency response of the
device.
It is also important to note that the Miller capacitance is the capacitance seen looking into the input.
If looking for all of the RC time constants (poles) it is important to include as well the capacitance
and
seen by the output. The capacitance on the output is often neglected since it sees
amplifier outputs are typically low impedance. However if the amplifier has a high impedance
output, such as if a gain stage is also the output stage, then this RC can have a significant impact on
the performance of the amplifier. This is when pole splitting techniques are used.
The Miller effect may also be exploited to synthesize larger capacitors from smaller ones. One such
example is in the stabilization of feedback amplifiers, where the required capacitance may be too
large to practically include in the circuit. This may be particularly important in the design of
integrated circuits, where capacitors can consume significant area, increasing costs.
Mitigation
The Miller effect may be undesired in many cases, and approaches may be sought to lower its
impact. Several such techniques are used in the design of amplifiers.
A current buffer stage may be added at the output to lower the gain
between the input and output
terminals of the amplifier (though not necessarily the overall gain). For example, a common base
may be used as a current buffer at the output of a common emitter stage, forming a cascode. This
will typically reduce the Miller effect and increase the bandwidth of the amplifier.
Alternatively, a voltage buffer may be used before the amplifier input, reducing the effective source
impedance seen by the input terminals. This lowers the
time constant of the circuit and typically
increases the bandwidth.
Impact on frequency response
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Figure 2A shows an example of Figure 1 where the
impedance coupling the input to the output is the
coupling capacitor CC. A Thévenin voltage source VA
drives the circuit with Thévenin resistance RA. The
output impedance of the amplifier is considered low
enough that the relationship Vo= AvVi is presumed to
hold. At the output ZL serves as the load. (The load is
irrelevant to this discussion: it just provides a path for
the current to leave the circuit.) In Figure 2A, the
coupling capacitor delivers a current jωCC(Vi − Vo) to the
output node.
Figure 2B shows a circuit electrically identical to Figure
2A using Miller's theorem. The coupling capacitor is
replaced on the input side of the circuit by the Miller
capacitance CM, which draws the same current from the
Figure 2: Amplifier with feedback
driver as the coupling capacitor in Figure 2A. Therefore,
capacitor CC.
the driver sees exactly the same loading in both circuits.
On the output side, a capacitor CMo = (1 + 1/Av)CC draws
the same current from the output as does the coupling capacitor in Figure 2A.
In order that the Miller capacitance draw the same current in Figure 2B as the coupling capacitor in
Figure 2A, the Miller transformation is used to relate CM to CC. In this example, this transformation
is equivalent to setting the currents equal, that is
or, rearranging this equation
This result is the same as CM of the Derivation Section.
The present example with Av frequency independent shows the implications of the Miller effect, and
therefore of CC, upon the frequency response of this circuit, and is typical of the impact of the Miller
effect (see, for example, common source). If CC = 0 F, the output voltage of the circuit is simply Av
vA, independent of frequency. However, when CC is not zero, Figure 2B shows the large Miller
capacitance appears at the input of the circuit. The voltage output of the circuit now becomes
and rolls off with frequency once frequency is high enough that ω CMRA ≥ 1. It is a low-pass filter. In
analog amplifiers this curtailment of frequency response is a major implication of the Miller effect.
In this example, the frequency ω3dB such that ω3dB CMRA = 1 marks the end of the low-frequency
response region and sets the bandwidth or cutoff frequency of the amplifier.
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It is important to notice that the effect of CM upon the amplifier bandwidth is greatly reduced for low
impedance drivers (CM RA is small if RA is small). Consequently, one way to minimize the Miller
effect upon bandwidth is to use a low-impedance driver, for example, by interposing a voltage
follower stage between the driver and the amplifier, which reduces the apparent driver impedance
seen by the amplifier.
The output voltage of this simple circuit is always Av vi. However, real amplifiers have output
resistance. If the amplifier output resistance is included in the analysis, the output voltage exhibits a
more complex frequency response and the impact of the frequency-dependent current source on the
output side must be taken into account.[3] Ordinarily these effects show up only at frequencies much
higher than the roll-off due to the Miller capacitance, so the analysis presented here is adequate to
determine the useful frequency range of an amplifier dominated by the Miller effect.
Miller approximation
This example also assumes Av is frequency independent, but more generally there is frequency
dependence of the amplifier contained implicitly in Av. Such frequency dependence of Av also makes
the Miller capacitance frequency dependent, so interpretation of CM as a capacitance becomes more
difficult. However, ordinarily any frequency dependence of Av arises only at frequencies much
higher than the roll-off with frequency caused by the Miller effect, so for frequencies up to the Miller
-effect roll-off of the gain, Av is accurately approximated by its low-frequency value. Determination
of CM using Av at low frequencies is the so-called Miller approximation.[2] With the Miller
approximation, CM becomes frequency independent, and its interpretation as a capacitance at low
frequencies is secure.
References and notes
1. ^ John M. Miller, "Dependence of the input impedance of a three-electrode vacuum tube upon the load in
the plate circuit," Scientific Papers of the Bureau of Standards, vol.15, no. 351, pages 367-385 (1920).
Available on-line at: http://web.mit.edu/klund/www/papers/jmiller.pdf .
2. ^ a b R.R. Spencer and M.S. Ghausi (2003). Introduction to electronic circuit design.
(http://worldcat.org/isbn/0-201-36183-3). Upper Saddle River NJ: Prentice Hall/Pearson Education, Inc.
p. 533. ISBN 0-201-36183-3.
3. ^ See article on pole splitting.
See also
■ Miller theorem
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